Optimizing (REF ) with \(\lambda =1\)  reduces to the original problem of selecting \(\psi \)  to minimize \({\rm KL}({\rm m}||{\rm p})\) . As long as the \({\rm q}(\mathbf {x};\theta )\)  component family can express Dirac-deltas at arbitrary \(\mu \) s, this guarantees at least one such unbiased \(\psi \) , namely the one equivalent to sampling. However, it may be the case that there are multiple such unbiased mixing distributions, e.g. if \({\rm p}(\mathbf {x})\)  itself is a mixture of components from the \({\rm q}-\) family; in the two-dimensional space of Mutual Information versus Expected KL, such a mixture would appear as a point on the \(y=x\)  line (Figure REF e). Maximizing the objective in (REF ) does not arbitrate among these different unbiased \(\psi \) s if there is more than one. However, the volume of \(\theta -\) space is larger in regions with large Fisher Information where mixture components are maximally distinguishable [1]}, i.e. where \(\theta \)  corresponds to deltas. Thus, we can conclude that if a process selects \(\theta \)  uniformly from among all unbiased \(\psi \) s, it is probable that \({\rm q}(\mathbf {x};\theta )\)  will be very sharp in the sense of high Fisher Information.
